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RESEARCH PRODUCT

Volumes transverses aux feuilletages d'efinissables dans des structures o-minimales

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Let Fλ be a family of codimension p foliations defined on a family Mλ of manifolds and let Xλ be a family of compact subsets of Mλ. Suppose that Fλ, Mλ and Xλ are definable in an o-minimal structure and that all leaves of Fλ are closed. Given a definable family Ωλ of differential p-forms satisfaying iZ Ωλ = 0 forany vector field Z tangent to Fλ, we prove that there exists a constant A > 0 such that the integral of on any transversal of Fλ intersecting each leaf in at most one point is bounded by A. We apply this result to prove that p-volumes of transverse sections of Fλ are uniformly bounded.